Page 84 - Singiresu S. Rao-Mechanical Vibrations in SI Units, Global Edition-Pearson (2017)

1.9 dampinG elements 81
Equations (E.9) and (E.8) lead to

2d
3
3pD l¢1 + ≤
C D S
P = mv 0 (E.10)
4d 3
By writing the force as P = cv 0 , the damping constant c can be found as
3
3pD l 2d
c = mJ ¢1 + ≤ R (E.11)
4d 3 D
■

1.9.2 If the force (F)-velocity (v) relationship of a damper is nonlinear:
linearization (1.26)
of a nonlinear F = F1v2
damper a linearization process can be used about the operating velocity 1v*2, as in the case of a
nonlinear spring. The linearization process gives the equivalent damping constant as

dF
c = 2 (1.27)
dv v*

1.9.3 In some dynamic systems, multiple dampers are used. In such cases, all the dampers are
Combination of replaced by a single equivalent damper. When dampers appear in combination, we can
dampers use procedures similar to those used in finding the equivalent spring constant of multiple
springs to find a single equivalent damper. For example, when two translational dampers,
with damping constants c and c , appear in combination, the equivalent damping constant
1
2
1c 2 can be found as (see Problem 1.55):
eq
Parallel dampers: c eq = c + c 2 (1.28)
1
1 1 1
Series dampers: = + (1.29)
c eq c 1 c 2

equivalent spring and damping Constants of a machine tool support
example 1.17
A precision milling machine is supported on four shock mounts, as shown in Fig. 1.45(a). The elas-
ticity and damping of each shock mount can be modeled as a spring and a viscous damper, as shown
in Fig. 1.45(b). Find the equivalent spring constant, k eq , and the equivalent damping constant, c eq ,
of the machine tool support in terms of the spring constants 1k i 2 and damping constants 1c i 2 of
the mounts.